= Perfect integral Hecke pairing
{title2=$\mathbb T\times S_k(\Gamma(1),\mathbb Z)\longrightarrow\mathbb Z$}
The pairing $(T,f)\mapsto a_1(Tf)$ identifies the <integral Hecke algebra of level-one cusp forms> with the integral dual of the cusp-form lattice. An integral basis beginning $q,q^2,\ldots,q^m$ gives a unitriangular first-$m$ coefficient matrix, so $a_1,\ldots,a_m$ form a dual basis. Commutativity of the <Hecke operators> proves nondegeneracy on the algebra, and $T_1,\ldots,T_m$ then form its integral basis.
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